Unbounded Holevo additivity gaps in finite dimensions

We establish unbounded two-use Holevo additivity gaps in finite dimensions. For each sufficiently large fixed integer K and all sufficiently large n, we construct channels T n with output dimension K n, input dimension (Θ K(n 2)), and [ χ(T n squared)-2χ(T n) n [2K over K-2 2(1+9/K)]-O(1/n).] The gap is linear in output qubits, with an explicit quadratic input-qubit cost and an explicit threshold on n. We also obtain channels whose single-use Holevo quantity tends to zero while their two-use Holevo information, and hence classical capacity, diverges. The channels arise from structured tensor products of the complementary mixed-unitary channels used in Collins's free-probabilistic proof. We establish minimum-output-entropy gaps by combining Collins--Youn's product-group Haagerup inequality with Bordenave--Collins's quantitative strong-convergence estimates. We also provide a Lean certificate for our proofs.